The sedimentation process in the hopper of a Trailing Suction Hopper
Dredge (TSHD) is very complex. However it is debatable whether for estimation purposes it
is necessary to model the processes involved in detail. For estimation purposes a black
box approach may suffice. Based on the turbulent diffusion theory of Camp (1936, 1946) and
Dobbins (1944) for sedimentation tanks, a method has been developed describing the
sedimentation and thus the overflow losses in hoppers. The Camp theory considers particles
of one diameter, but here the grain distribution is considered, which is the basis for
calculating the overflow losses. In the time domain, a homogeneous flow above the sand bed
is assumed. Using Camp as a first criterion, it is possible to determine which percentage
of particles of a specified diameter will settle and which particles will have too small a
settling velocity to settle. A second criterion is the scour velocity. If the velocity
above the bed is greater than the scour velocity of a particle of a specified diameter,
that particle will be re suspended. Both criteria will result in overflow losses. Given a
specified mixture flow into the hopper, once the overflow losses are known, the loading
curve can be determined. Since non-linearities are involved, the calculations are carried
out in the time domain.

The model is implemented in a computer program, which also calculates
the stratification of the grain distribution in the hopper and the grain distribution of
the material leaving the hopper through the overflow as a function of time.

This paper is a follow-up of the paper of Vlasblom and Miedema (1995) at
WODCON XIV. Some of the equations have been modified resulting in different loading
curves. The implementation of scour has changed and a discussion about two-dimensional
turbulent diffusion models has been added.

The sedimentation process in TSHD's has not been the subject for many
publications. Although research has been carried out, there is not a good prediction model
available. Models that exists are based on sediment flow in rivers or on sedimentation
tanks. Camp (1946) and Dobbins (1944) developed a theory for ideal settling tanks. In
these tanks an entrance zone, a settling zone and an overflow zone can be distinguished.
Settled grains will be removed periodically, so the tank will not be filled until the
overflow level is reached as with TSHD's. Another difference with TSHD's is that the
entrance and the overflow zone are not used for the settling process. In the hopper of a
TSHD, although an entrance and an overflow zone exist, the hopper as a whole will be used
for the settling process. In the hopper of a TSHD the flow velocity above the sediment is
much higher then it is in a settling tank. With a few adaptations however the Camp and
Dobbins theory can be applied to TSHD's. These adaptations are:

- The hopper as a whole is used for the sedimentation process.

- The sedimentation process continues until the hopper is economically
full.

- The overflow level may change during the sedimentation process.

- A grain size distribution will be used.

To understand the adapted model, first the settling of grains will be
considered. With this knowledge the ideal settling basin will be discussed. Before
implementing the Camp and Dobbins theory to a TSHD, the loading cycle of the TSHD will be
explained. With some case studies the use of the model as applied to TSHD's will be
discussed.

The settling velocity of grains depends on the grain size, shape and
specific density. It also depends on the density and the viscosity of the fluid the grains
are settling in, it also depends upon whether the settling process is laminar or
turbulent. In general, the settling velocity v can be determined with the following
equation:

(1)

The Reynolds number of the settling process determines whether the
process is laminar or turbulent. The Reynolds number can be determined by:

(2)

The drag coefficient Cd depends upon the Reynolds
number according to:

(3)

(4)

(5)

Stokes, Budryck and Rittinger used these drag coefficients to calculate
settling velocities for laminar settling (Stokes), a transition zone (Budryck) and
turbulent settling (Rittinger) of sand grains. This gives the following equations for the
settling velocity:

Laminar flow, d<0.1 mm, according to Stokes.

(6)

Transition zone, d>0.1 mm and d<1 mm, according to
Budryck.

(7)

Turbulent flow, d>1, according to Rittinger.

(8)

In these equations the grain diameter is in mm and the settling velocity
in mm/sec. Since the equations were derived for sand grains, the shape factor for sand
grains is used for determining the constants in the equations. The shape factor can be
introduced into the equations for the drag coefficient by dividing the drag coefficient by
a shape factor y . For normal
sands this shape factor has a value of 0.7. The viscosity of the water is temperature
dependent. If a temperature of 10° is used as a reference, then the viscosity increases by 27% at 0° and it decreases by 30% at 20° centigrade. Since the viscosity influences
the Reynolds number, the settling velocity for laminar settling is also influenced by the
viscosity. For turbulent settling the drag coefficient does not depend on the Reynolds
number, so this settling process is not influenced by the viscosity. Other researchers use
slightly different constants in these equations but, these equations suffice to explain
the basics of the settling process in hopper dredges.

The above equations calculate the settling velocities for individual
grains. The grain moves downwards and the same volume of water has to move upwards. In a
mixture, this means that, when many grains are settling, an average upwards velocity of
the water exists. This results in a decrease of the settling velocity, which is often
referred to as hindered settling. However, at very low concentrations the settling
velocity will increase because the grains settle in each others shadow. Richardson and
Zaki determined an equation to calculate the influence of hindered settling for volume
concentrations Cv between 0 and 0.3. The coefficient in this equation is
dependent on the Reynolds number.

The ideal settlement basin (Fig. 1 & 2) consists of an entrance zone
where the solid/fluid mixture enters the basin and where the grain distribution is uniform
over the cross-section of the basin, a settlement zone where the grains settle into a
sediment zone and a zone where the cleared water leaves the basin, the overflow zone. It
is assumed that the grains are distributed uniformly and are extracted from the flow when
the sediment zone is reached. Each particle stays in the basin for a fixed time and moves
from the position at the entrance zone, where it enters the basin towards the sediment
zone, following a straight line. The slope of this line depends on the settling velocity v
and the flow velocity above the sediment so. Figure 1 shows a top view
of the ideal settlement basin. Figure 2 shows the side view and Figure 3 the path of
individual grains. All particles with a diameter d0 will settle, if a
particle with this diameter entering the basin at the top, reaches the end of the sediment
zone. Particles with a larger diameter will all settle, particles with a smaller diameter
will partially settle. The settling velocity of a grain with diameter d0
can be determined by:

(10)

With:

(11)

The settling velocity vo is often referred to as the
hopper load parameter. A small hopper load parameter means that small grains will settle.
From figure 3 the conclusion can be drawn that grains with a settling velocity greater
then vo will all reach the sediment layer and thus have a settling
efficiency of 1. Grains with a settling velocity smaller then vo will
only settle in the sedimentation zone, if they enter the basin below a specified level.
These grains have a settling efficiency of v/vo. If the fraction of
grains with a settling velocity

greater then vo equals po, then the
settling efficiency for a grain distribution can be determined by integrating the grain
settling efficiency:

(12)

(13)

This is illustrated in figure 5, showing a grain distribution curve, but
instead of using the grain diameter on the horizontal axis, the settling velocity divided
by the hopper load parameter is used. The hatched area is equal to the total settling
efficiency r. Until now the flow velocity distribution has been considered uniform.
If this distribution is not uniform, as shown in figure 4, it can be shown that the
settling efficiency does not change (Camp 1946, de Koning 1977). For the ideal settlement
basin laminar flow is assumed. Turbulent flow will reduce the settling velocity of the
grains and thus the total settling efficiency. Whether turbulent flow occurs, depends on
the Reynolds number of the flow in the basin. Using the hydraulic diameter concept this
number is:

(14)

For a given flow Q and viscosity n the Reynolds number depends on the width W
and the height H of the basin. A large width and height give a low Reynolds number.
However this does not give an attractive shape for the basin from an economical point of
view, which explains why the flow will be turbulent in existing basins.

Dobbins (1944) and Camp (1944, 1946) use the two-dimensional turbulent
diffusion equation to determine the resulting decrease of the settling efficiency.

(15)

Assuming a parabolic velocity distribution instead of the logaritmic
distribution, neglecting diffusion in the x-direction and considering the settling
velocity independent of the concentration reduces the equation to:

(16)

Because of the parabolic velocity distribution, the turbulent diffusion
coefficient ez
is a constant. A further simplification is obtained if the velocity s is assumed
constant throughout the depth, meaning that the constant of the parabola k approaches
zero. In this case the turbulent diffusion equation becomes:

(17)

Huisman (1995) in his lecture notes, derives the diffusion-dispersion
equation in a more general form, including longitudinal dispersion.

(18)

Assuming a steady and uniform flow, the longitudinal disperion
coefficient is independent of x and the settling velocity v independent of z.
This reduces the equation 18 to:

(19)

By means of computations Huisman (1995) shows that the retarding effect
of dispersion may be ignored for the commonly applied width to depth ratio 3 to 5. This
reduces equation 19 to equation 17 of Dobbins and Camp.

Groot (1981) investigated the influence of hindered settling and the
influence of different velocity distributions using the following equation:

(20)

The velocity distribution, the diffusion coefficient distribution and
the distribution of the initial concentration did not have a significant influence on the
computed results, but the results were very sensitive on the formulation of hindered
settling. This formulation of course influences the settling velocity in general.

Equation 17 can be solved analytically using separation of variables.
The boundary conditions used by Camp and Dobbins describe the rate of vertical transport
across the water surface and the sediment for x=¥ and the concentration distribution at the inlet, these are:

at the water surface

at the sediment for x=¥ , for the no-scour situation

at the entrance for x=0

This method, resulting in figure 6, gives the removal ration due to
turbulence for a single grain. The removal ratio can be determined by summation of a
series. Solving equation 17 gives (v× H/2×ez) as the independent
parameter on the horizontal axis and the removal ratio (v/vo=settling
efficiency) on the vertical axis. Using a parabolic velocity distribution this can be
substituted by:

(21)

with: k = 0.4
and l = 0.03

It is however of interest how the removal ratio (settling efficiency) of
a grain can be split up in a part determined by laminar flow in the basin according to
equation (12) and a part caused by turbulence. Vlasblom and Miedema (1995) give relations
for the effect of turbulence only (fig. 7), as derived by Miedema (1991):

(22)

For values of v/vo greater then 1 the following equation should
be used.

(23)

Equations 22 and 23 have been revised since Vlasblom and Miedema (1995)
to match the results of equation 17 more closely. For other values of the viscous friction
factor l , the constant .5
should be taken as: .5 + .5×
Log(0.03/l )

The resulting basin settling efficiency is equal to the grain settling
efficiency times the turbulence settling efficiency, according to figure 6. The total
settling efficiency for the basin can now be determined by:

(24)

When the height of the sediment increases and the hopper load parameter
remains constant, the horizontal flow velocity above the sediment also increases. Grains
that have already settled will be re suspended and leave the basin through the overflow.
This is called scouring. The scour velocity ss for a grain with a
diameter ds is according to Camp (1946):

(25)

Grains that are re suspended due to scour, will not stay in the basin
and thus have a settling efficiency of zero. In this equation l is the viscous friction coefficient mobilized on the
top surface of the sediment and has a value in the range of 0.01-0.03, depending upon the
Reynolds number and the ratio between the hydraulic diameter and the grain size (surface
roughness). The value of l
should be choosen equal to the value in equation 21. The porosity n has a value in
the range 0.6-0.4, while the friction coefficient m depends on the internal friction of the sediment and has a value in the range of
0.1-0.6 for sand grains. To prevent scour for a specified grain diameter, the flow
velocity has to be smaller then the scour velocity. The total settling efficiency changes
due to scour, according to:

(26)

If the settling process does not occur in some parts of the basin, owing
to a concentrated inflow of mixture, vortexes, etc. the effective hopper load parameter
will increase and the settling efficiency will decrease. The homogeneity of the flow
depends on the stability of the flow. The stability is indicated by the Froude number of
the basin. This Froude number is:

(27)

A higher Froude number indicates higher stability to the flow in the
basin. This results in a narrow basin with a small height. With respect to turbulence the
demand for a high Froude number conflicts with the demand for a small Reynolds number,
which is associated with a wide deep basin.

The loading cycle is considered to start when the hopper is filled with
soil and starts to sail to the dump area. This point in the loading cycle was chosen as
the starting point to show the optimal load in a graph. The loading cycle then consists of
the following phases:

Phase 1: The water above the overflow level flows away through the
overflow. The overflow is lowered to the sediment level, so the water above the sediment
can also flow away. In this way minimum draught is achieved. Sailing to the dump area is
started.

Phase 2: Continue sailing to the dump area.

Phase 3: Dump the load in the dump area.

Phase 4: Pump the remaining water out of the hopper and sail to the
dredging area.

Phase 5: Start dredging and fill the hopper with mixture to the overflow
level, during this phase 100% of the soil is assumed to settle in the hopper.

Phase 6: Continue loading with minimum overflow losses, during this
phase, according to equation 26, a percentage of the grains will settle in the hopper. The
percentage depends on the grain size distribution of the sand.

Phase 7: The maximum draught (CTS, Constant Tonnage System) is reached.
from this point on the overflow is lowered. Equation 26 is still valid.

Phase 8: The sediment in the hopper is rising due to sedimentation, the
flow velocity above the sediment increases, resulting in scour. This is the cause of
rapidly increasing overflow losses.

These phases are shown in fig. 8. The way each phase occurs in the
cycle, depends on the type of hopper dredge, the working method and of course, the type of
soil to be dredged.

The sedimentation in the hopper occurs during the phases 5, 6, 7 and 8.
During phase 5 the hopper is filled with mixture until the overflow level is reached.
During this phase 100% of the soil is assumed to stay in the hopper and settle. When the
overflow level is reached, phase 6, depending on the grain distribution, a specified
percentage of the soil will not settle and will leave the hopper via the overflow. During
this phase scouring does not have much influence on the sedimentation process. When the
maximum weight of the hopper contents is reached, the overflow will be lowered
continuously in order to keep the weight of the hopper contents constant at its maximum.
When the sediment level rises, phase 8, the flow velocity above the sediment increases and
scouring will re suspend settled particles. The overflow losses increase with time. The
transition between phase 5 and 6 is very sharp, as is the transition between the phases 6
and 7 for the graph of the total load, but this does not exist in the graph of the
effective load (fig. 8). However, the transition between the phases 7 and 8 is not
necessarily very sharp. When this transition occurs depends on the grain distribution of
the soil dredged. With very fine sands this transition will be near the transition between
phases 6 and 7, so phase 7 is very short or may not occur at all. With very coarse sands
and gravel scouring is minimal, so phase 8 is hardly present. In this case the sediment
level may be higher then the overflow level. With silt the phases 7 and 8 will not occur,
since after reaching the overflow level the overflow losses will be 100%.

So far the total load in the hopper has been described. A contractor is,
of course, interested in the "Tonnes Dry Solids" (TDS) or situ cubic meters. The
total load or gross load consists of the sediment with water in the pores and a layer of
water above the sediment. The TDS consists of the weight of the soil grains only. The net
weight in the hopper consists of the weight of the sediment, including the weight of the
pore water. If the porosity of the sediment is considered to be equal to the in-situ
porosity, then the volume of the sediment in the hopper equals the removed situ-volume.
Although, in practice, there will be a difference between the in-situ porosity and the
sediment porosity, here they will be considered equal. The net weight can be calculated
according to:

(28)

(29)

(30)

(31)

(32)

(33)

(34)

(35)

The net weight (situ weight) according to equation 34 can be
approximated by the total weight of the load in the hopper minus the weight of the same
volume of water and the result multiplied by 2. For the TDS this factor is about 1.2,
according to equation 35. This is of course only valid for a specific density of the
sediment of 2 tons per cubic meter.

With these equations the hopper cycle for the net weight and the TDS can
be derived, this is shown in the figures 8 to 11. The hopper dredge is optimally loaded,
when the effective load (weight) or the TDS divided by the total cycle time dWs/dt
reaches its maximum. This is shown in figures 8 to 11 and is the reason for the starting
point of the loading cycle in figure 8.

The previous paragraphs explain the basics of the settling velocity of
grains, the ideal settlement basin and the loading cycle of a hopper dredge. These basic
theories will now be applied to the sedimentation process in a hopper.

Consider a rectangular hopper of width W, height H and
length L. A mixture with a mixture density rm and with a specified grain distribution is being dredged.
Depending on the operational conditions such as dredging depth, the pump system installed
and the grain distribution and mixture density, a mixture flow Q will enter the
hopper. If the porosity n of the sediment is known, the flow of sediment can be
determined according to:

The mass flow of the mixture is:

(36)

The mass flow of the solids is now:

(37)

From this, the mass flow of sediment is:

(38)

With:

(39)

Part of this mass flow will settle in the hopper and another part will
leave the hopper through the overflow. The ratio between these parts depends on the phase
of the loading process. During phase 5 the hopper is loaded to the overflow level, so the
mass flow into the hopper will stay in the hopper. This means that the total settling
efficiency during this phase equals 1. During phase 6 the loading continues until the
maximum load in the hopper is reached. If scouring does not occur, the mass flow that will
settle into the sediment can be calculated with equation 38, where the settling efficiency
should be determined with equation 24.

During phase 7 the loading continues, but with a CTS, the overflow is
lowered to ensure that the total weight in the hopper remains constant. As scour does not
yet occur, the above equation is still valid. During phase 8 scouring occurs. If scouring
does occur, the mass flow that will settle into the sediment can be calculated also with
equation 38, but the settling efficiency should be determined with equation 26. Scouring
is the cause of increasing overflow losses. Scour depends upon the velocity of the flow
above the sediment. Since in a hopper the sediment is not removed, the sediment level
rises during the loading of the hopper. This means that the height of the mixture flow
above the sediment decreases during the loading process, resulting in an increasing flow
velocity. The scour velocity can now be determined by:

(40)

With:

(41)

The height H is a constant for a Constant Volume System (CVS),
but this height changes for a CTS, because the overflow is lowered from the moment, the
maximum weight in the hopper is reached. If a maximum weight Wm is
considered, the height of the layer of water above the sediment for a CTS can be
determined by:

(42)

The hopper loading curve can now be determined by first calculating the
time required to fill the hopper (phase 6), given a specified mixture flow Q. From
the mixture density rm
the mass and given a specified porosity, the volume of the sediment can be calculated.
From this point the calculations are carried out in small time steps (phases 7 and 8). In
one time step, first the height of the sediment and the height of the water layer above
the sediment are determined. The height of the water layer can be determined with equation
41 for a CVS hopper and equation 42 for a CTS hopper. With equation 40 the scour velocity
can now be determined. Using equation 25 the fraction of the grains that will be subject
to scour can be determined. If this fraction ps is zero equation 24 has
to be used to determine the mass flow that will stay in the hopper. If this fraction is
not equal to zero equation 26 has to be used. Equation 38 can now be used to determine the
mass flow. This mass flow multiplied by the time step results in an increment of the
sediment mass that is added to the already existing mass of the sediment. The total
sediment mass is the starting point for the next time step. This is repeated until the
overflow losses are 100%. When the whole loading curve is known, the optimum loading time
can be determined. This is shown in Figure 8, where the dotted line just hits the loading
curve of the effective (situ) load. The point determined in this way gives the maximum
ratio of effective load in the hopper and total cycle time.

As stated before, the ideal settlement basin is a rectangular basin with
an entrance zone, a settlement and sedimentation zone and a overflow zone. The hopper
geometry and configuration aboard of the TSHD can be quite different from the ideal
situation, so a method to schematise the hopper dimensions is required.

1. The height H of the hopper can be defined best as the hopper
volume divided by the hopper area L×W. This means that the base of the ideal hopper, related to the maximum
overflow height is at a higher level than the ship's base. This assumption results in a
good approximation at the final phases (7 and 8) of the loading process, while in phase 6
of the loading process the hopper is filled with mixture and so the material stays in the
hopper anyway.

2. Near the loading chute of the hopper or in cases where a deep loading
system is used, the turbulence of the flow results in a good and sufficient distribution
of the concentration and particle size distribution over the cross-section of the hopper,
so the entrance zone can be kept small. For example between the hopper bulkhead and the
end of the loading chute.

3. In the ideal settlement basin there are no vertical flow velocities
except those resulting from turbulence. However in reality vertical velocities do occur
near the overflow, therefore it is assumed that the overflow zone starts where the
vertical velocities exceed the horizontal velocities. An estimate of where this will occur
can easily be made with a flow net.

4. Although the presence of beams and cylinder rods for the hopper doors
does increase the turbulence, it is the authors opinion, that an additional allowance is
not required, neither for the hopper load parameter, nor for the turbulence parameter.

To give an impression of the behaviour and the sensitivity of the Camp
and Dobbins model, three cases are calculated with the computer program TSHD (Miedema
1991). The rectangular hopper used, has a length L, of 46.3 m, a width W, of
9.8 m and a height H, of 6.0 m. At the entrance zone the mixture enters the hopper
at a flow rate of approximately 4.6 m3/sec and a mixture density of 1.25 tons/m3.
The hopper has a design density of 1.48 tons/m3. The hopper loading cycle
consists of sailing to the dump area (phase 1 + 2); 120 min., dumping (phase 3); 15 min.,
sailing back to the dredging area (phase 4); 120 min., filling the hopper up to the
overflow level (phase 5); about 10 min. and continue loading until the optimum loading
cycle production is reached (phase 6, 7 and 8). The hopper is of the CTS type (de Koning
1977), so the overflow will be lowered when the maximum weight is reached.

The flow rate and mixture density are chosen relatively high to
emphasise the overflow losses.

The calculations are carried out with three sands. A sand with a d50
of 100 mm (sand A), a sand with
a d50 of 300 mm (sand
B) and a sand with a d50 of 1000 mm (sand C).

The grain distributions are determined by integration of a Gauss
distribution, where the d50 is equivalent to the mean value and (d50-d85)
is equivalent to the variance of the Gauss distribution. The cumulative grain distribution
is determined at 100 grain diameters. This way the total settling efficiency, according to
equation 25, can be determined accurate enough. For each grain diameter at each time step,
the fraction that settles and thus the fraction that leaves the hopper through the
overflow can be determined. The result of this is, that the cumulative grain distribution
of the sand at the top of the sedimentation zone and of the sand leaving through the
overflow can be generated.

For determining the scour velocity, a porosity n of 0.4, a
friction coefficient m of 0.577
(30° ) and a viscous friction
coefficient l of 0.03 are used.

For sand A the total loading cycle is shown in figure 8. Figure 9 is a
close-up of the phases 5, 6, 7 and 8. Figure 12 shows the original grain distribution (the
thick solid line in the middle), at the left of the original distribution, the grain
distributions of the sand leaving the hopper at time intervals of 5 min. and at the right
of the original distribution, the grain distributions at the top of the sedimentation
zone, also at intervals of 5 min. The dotted line in figures 8 and 9 shows optimal
loading. In the graphs the loading is continued for several minutes after this optimal
point to emphasise the overflow losses. Figures 10 and 13 give the same information for
sand B and Figures 11 and 14 give this information for sand C.

These figures show that the optimal loading time decreases with an
increasing grain diameter. The overflow losses also decrease with an increasing grain
diameter, which is evident. The tendencies as they are calculated match practice, however
the model has to be tuned to every specific dredge to be a tool for estimating purposes.

The Camp and Dobbins model can be used to estimate loading time and
overflow losses, however, the model should be tuned with measurements of the overflow rate
in tons/sec as well as the particle size distribution in the overflow, as a function of
time. The model can then also be used for the calculation of the decaying of the overflow
plume in the dredging area.

If the model is used for the calculation of the production rate of the
dredge a distinction has to be made whether the production is expressed in T.D.S./sec or
in m3/sec. In the first case the theory can be applied directly, while in the
second case it has to be realised, that the overflow losses in T.D.S./sec do not always
result in the same overflow loss in m3/sec, since fine particles may situate in
the voids of the bigger ones. The loss of fines does not reduce the total volume, but
increases the void ratio. Although the fines leave the hopper in this case, they do not
result in a reduction of the volume of the settled grains.

Those fractions which can be considered to apply to the overflow losses
and those which do not, can be estimated from the difference between the real particle
size distribution and the optimal particle size distribution, giving a maximum dry
density, the so called Fuller distribution. If the gradient of the distribution curve for
the fines is less steep then the corresponding gradient of the Fuller distribution, than
that fraction of fines will not effectively contribute to the overflow losses if they are
expressed in m3/sec. In such a case, in-situ, the fines were situated in the voids of the
courser grains. If the gradient is

however steeper, the fines also form the grain matrix and the volume of
settled grains will decrease if the fines leave the hopper through the overflow. Figure 15
gives an example of the Fuller distribution compared with a real grain distribution.

In the model a number of assumptions are made. Except from numerical
values for the parameters involved, the Camp and Dobbins approach is used for the
influence of turbulence, while seperately the influence of scour is used instead of using
it as a boundary condition. During phase 8 of the loading process scour dominates the
overflow losses. This of course depends on the way scour is modelled. In the case studies
equation 25 according to Camp (1946) is used. Whether this is correct will be one of the
subjects of further research.

Until now it is very difficult to get correct data from the field to
verify this theory. It is not only necessary to have loading curves, but also grain
distributions of the sediment and in the overflow, which are hard to obtain.